Propagation rate of large cracks in stiffened panels under tension loading

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Marine Structures 18 (2005) 265–288 www.elsevier.com/locate/marstruc

Propagation rate of large cracks in stiffened panels under tension loading Hussam N. Mahmouda,, Robert J. Dexterb,{ a

ATLSS Research Center, Lehigh University, 117 ATLSS Dr., Bethlehem, PA 18018, USA Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455, USA

b

Received 15 December 2004; received in revised form 7 August 2005; accepted 6 September 2005

Abstract Cyclic tension fatigue tests were conducted on approximately half-scale welded stiffened panels to study propagation of large cracks as they interact with the stiffeners. A linear elastic fracture mechanics analysis was used, to simulate the crack propagation, and gave reasonable agreement with the experiments. The range in stress intensity factor (DK) was determined with either a finite-element (FE) analysis or an analytical model at increments of crack length. Analytical and FE models included an idealized residual stress distribution similar to what was measured in the panels. Crack propagation rate as a function of DK was estimated using the Paris law with upper-bound coefficients. The experiments and analyses show little sensitivity to stiffener type. The models developed in this project can be easily reproduced and can be used to assess the remaining life of ships with large cracks, leading to more accurate assessment of safety and more efficient scheduling of repairs. r 2005 Elsevier Ltd. All rights reserved. Keywords: Long cracks; Crack propagation; Residual stresses; Stress intensity factor

1. Introduction Generally, crack-like discontinuities or flaws exist in structural elements due to welding. Under cyclic loading, a flaw can develop into a fatigue crack and propagate until fracture Corresponding author. Tel.: +1 610 758 3066; fax: +1 610 758 3555.

E-mail address: [email protected] (H.N. Mahmoud). The Late.

{

0951-8339/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.marstruc.2005.09.001

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occurs. Fracture mechanics is used to analyze and predict the behavior of a cracked structural element. Ships are very redundant and cracks are usually not an immediate threat to the integrity of the ship. Fatigue cracks in ships could grow to be as large as 24 m as shown in Fig. 1. These long cracks remain surprisingly stable because of minimum notch-toughness levels specified for ship steel and the redundancy of the structure. However, even if brittle fracture does not occur, at some point a net-section plastic collapse of the structure will occur, leading to a failure of the entire ship. A large tanker may have hundreds or even thousands of fatigue cracks discovered during inspection. These cracks are usually not an immediate threat to the structural integrity of the ship. Guidelines prepared by Tanker Structure Co-operative Forum are used for the inspection and maintenance of tanker structures [2,3]. The tolerance of ships to these cracks is a function of the overall structural redundancy and ductility, as well as fracture toughness of the structural components. The S2N curve approach is the most common method used for determining the fatigue life of a structural detail. The S2N curve is a lower bound to fatigue test data in terms of the stress range (S) and number of cycles to failure (N). Failure in the tests is usually defined as the development of through-thickness cracks (typically about 100 mm in length). Although the 24 m crack shown in Fig. 1 is an extreme case, it is not unusual to find cracks several meters long in a ship. The additional life for a through-thickness crack about 100 mm in length to grow to a length of several meters is quite substantial. Furthermore, if a high level of redundancy, ductility, and fracture toughness exist in the structure, the growth could be stable giving additional time to more efficiently schedule repair and maintenance.

24 m

Fig. 1. Cracked deck in the tanker Castor [1].

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1.1. Research objective Several studies on cracks in riveted aircraft stiffened panels have been conducted. However, few studies on crack growth in panels with welded stiffeners have been performed. Furthermore, few studies on crack growth in plates with welded stiffeners have been conducted where the panels are loaded axially. Welded stiffeners affect crack growth because of residual stresses present from the welding process. The high heat input from the welding process causes high tensile stresses in the vicinity of the stiffeners. The tensile stresses are equilibrated by compressive stresses in the region between the stiffeners. These residual stresses decrease the crack growth rate between stiffeners. Depending on the magnitude of the compressive residual stresses and the stress ranges, the crack growth can even arrest completely. Previous studies were done on box sections with edge webs and/or multiple stiffeners. The results showed stable propagation of the crack [4,5]. It was believed that a main factor in the stability of the growth was redundancy. Therefore, testing a single plate with no redundancy was essential in quantifying the effect of redundancy on the crack growth rate. The models developed in this research are aimed at predicting the growth of large cracks after they become through-thickness cracks, and investigating the effect of residual stresses and redundancy on crack growth. 2. Background 2.1. Fracture mechanics The theory of fracture mechanics can be used to predict fracture as well as fatigue crack propagation. Linear elastic fracture mechanics (LEFM) can be used under conditions where plastic deformation around the crack tip is relatively small. LEFM is applicable to high-cycle fatigue crack growth, which typically occurs when applied stresses are well below the yield stress of the steel. It is based on a parameter called the stress-intensity factor (K), which characterizes the intensity of the stress surrounding a sharp crack tip in a linear elastic and isotropic material [6]. 2.2. Modeling of fatigue crack growth Various models that were developed and used to predict fatigue crack growth are described by Rushton [7]. Most of the models have the form of a power law, which is associated with curve-fitting parameters that do not have a physical significance. One of the most reliable and effective models used for predicting fatigue crack growth is the Paris model, also known as the Paris Law. Paris and Erdogan [8] hypothesized in 1963 that the range in stress-intensity factor, DK, governs fatigue crack growth. The empirical Paris Law represents the crack growth rate data as a straight-line when plotted on a log–log scale. However, experimental da/dN verses DK data typically exhibit a sigmoid shape when plotted on a log–log scale. There is a DK threshold, DKth, below which cracks will not propagate. DKth can be taken as 3 MPam1/2 for structural steel. The Paris Law is fit to the linear portion of the da/dN versus DK plot (on a log–log scale) that lies above DKth. At relatively high DK levels, the crack growth rate

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accelerates and is accompanied by ductile tearing or increments of brittle fracture in each cycle. The Paris Law is expressed as da ¼ C
ðDK eff Þm , dN

(1)

where a is the half the crack length, N the number of cycles, C is an experimentally determined coefficient, DK the stress intensity factor range, m the material constant. It is difficult to achieve great accuracy in predicting crack growth rate when using fracture mechanics [9]. There is up to a factor of 20 in the scatter of experimental da/dN data. A great deal of the scatter is due to experimental error, especially at low growth rates near the threshold. In this range, the growth rate is affected by the procedure used to precrack the specimens at higher DK. There is a great deal of inherent variability in the actual growth rates, even if they were to be accurately measured. Crack growth rate depends on the load ratio (R) as indicated in Eq. (2). Different lines may be fit to the experimental data (on a log–log plot) for different load ratios. There are also empirical equations that account for the effect of the load ratio on the crack growth rate. The preferred approach is to account for the effect of load ratio and associated crack closure in the definition of DK. The idea is that the baseline crack growth rate model is defined for high load ratios (greater than 0.7), for which there is negligible crack closure. At lower load ratios (less than 0.5) and negative load ratios the crack is closed for part of the load cycle, and only the part of the load cycle where the crack is open is effective. An effective DKeff for a defined crack length is defined as that part of DK that contributes to the crack propagation and is computed by superposition of the values of Kapp, max, Kapp, min, and Kres, where Kapp, max is the magnitude of the stress intensity factor for a given crack length under the maximum applied load, Kapp, min is the magnitude of the stress intensity factor for a given crack length under the minimum applied load, and Kres is the magnitude of the stress intensity factor for a given crack length under the effect of residual stresses. DKeff is used in Eq. (1), where the parameters of Eq. (1) are defined by fitting the equation to high R crack growth rate data. This is the approach taken in this research, and the Paris Law will be assumed to be defined for high load ratio (greater than 0.7) R¼

smin , smax

(2)

where R is the Load ratio, smin the minimum applied stress, smax the maximum applied stress. Variance in the crack growth rate is usually expressed by variance in the coefficient C. Most researchers agree that all C–Mn steels have similar crack growth rates, and that the variance observed is just the typical material variation. In other words, there is no significant difference in the crack growth rates among various types of C–Mn steel, there is only scatter. Therefore, most reported values of C are intended to represent a conservative upper bound to the data. Barsom and Rolfe [10] established an upper bound for a variety of ferritic steels where C was 6.8  1012 for units of MPa and meters. The Barsom and Rolfe relation seems to be unconservative for high-load-ratio crack growth rates, however. Fisher et al. [12] performed a study of HSLA-80 steel, which showed that the upper

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bound value of C for high-load-ratio data was 9.0  1012. British Standards Institute BS 7910 [11] recommends an upper bound of 16.5  1012 for C. However, this value seems to be excessively conservative. In a previous version of this document known as PD 6493 [13], a more reasonable upper bound of 9.5  1012 was recommended for C.

2.3. Crack propagation in stiffened panels There has been extensive previous research on the solution for the stress intensity factor for a crack propagating in a stiffened panel. The majority of the work was done to investigate the growth of a crack in an aluminum panel for application to the aviation industry. Poe [14] proposed a numerical solution for K in a cracked panel with uniformly spaced riveted stiffeners. Poe introduced a parameter called ‘‘percent stiffening’’, which compares the area of the stiffeners to the total area of the stiffeners and the plate. He introduced this ratio and defined it as m. Poe [15] developed a solution for a crack propagating in a stiffened plate where the stiffeners were attached to the plate by means of rivets, and noticed that the K solution decreases as the crack approaches a stiffener, indicating that the stiffener aided in restraining the crack or slowing down the propagation. Poe also realized that the riveted stiffeners continue to limit crack growth after the crack propagates past the stiffener since a crack cannot propagate directly up into the stiffener. In a welded stiffener the crack may, however, propagate into and completely sever the stiffener. For integral stiffeners such as welded stiffeners, Poe developed a solution that assumed that once a crack reaches a stiffener, the stiffener is completely and suddenly severed and the load previously carried by the stiffener is shed to the remaining net section. Fig. 2 depicts this solution, which shows the jump in the K value as a result of the stiffener being immediately severed. Because there are only localized residual stresses induced in a plate with riveted connections and integrated stiffeners by extrusion, the study did not include the effect of residual stresses. The K value obtained from Poe’s model along with the Paris Law for crack propagation became a solid foundation for crack propagation in aircraft panels. Thayamballi [16] studied the effect of residual stresses on crack propagation in welded stiffened panels and outlined an analytical approach to calculate the fatigue crack growth. The contribution of the residual stresses to K was based on Greene’s function, integrating the solution for a pair of forces acting on the crack faces. A block tension and compression region, which was suggested by Faulkner [17], was used for the residual stress distribution. Using a three-flanged box beam to simulate the structural redundancy found in a doubled-hulled ship structure, Nussbaumer [4] performed a study on crack propagation in large-scale experiments on welded box girders. He also investigated the effect of residual stresses on the propagation of the crack using finite-element (FE) analysis and applying fictitious temperature fields to give thermal stresses in reasonable agreement with the initial residual stresses measured in the box girder. There was significant scatter in the measured residual stress distributions for these box girders. Therefore, three distributions were examined for the FE modeling based on the smallest, largest, and average observed residual stresses measured in the specimen.

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2s

ast

a 1.6

1

K

σ(πa) 2

1.4

1.2

1.0

0.8 0.5

1.0

1.5

2.0

a/2s Fig. 2. K solution for a panel with integral stiffeners.

Nussbaumer showed that the results were quite sensitive to the magnitude of residual stresses used for the analysis. Dexter and Pilarski [5] further extended Nussbaumer’s work to the case of multiple stiffener plate geometry rather than the un-stiffened cellular geometry. They tested box sections with drain holes, raised drain holes, and weld access holes. Their analyses also showed that the results were sensitive to the magnitude of residual stresses. 3. Description of experiment 3.1. Test set-up The test fixture was specially developed that uses leverage to reach up to 2400 kN load on the specimen from a 480 kN actuator. Fig. 3 shows the test fixture with a specimen in place. As shown in Fig. 4, the test fixture was a frame. The top and the bottom of the frame each consisted of four W27  84 beams. The top beams as well as the bottom beams were connected using diaphragms. These diaphragms ensured that all beams would experience the same displacement, resulting in an evenly distributed load across the tested panel. The left side of the test frame was an actuator that was used to introduce the cyclic loading and to create the tensile stresses on the specimen. The actuator was connected to the top and bottom beams using W14  132 spreader beams (one on the top and one on the bottom). Four columns, W12  50, were placed at the right side of the frame to support the

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Fig. 3. Test frame with specimen in place.

top four beams and to create a point of rotation that allows for the fatigue cycles. This point of rotation was introduced by placing a pin connection on the top of the columns. The bottom beams and the columns were fillet welded together. Finally, the test specimen was placed between the actuator and the columns at a distance of approximately 2440 mm away from the actuator or 610 mm away from the columns, and connected to the top and bottom beams with high-strength structural bolts, using double angles. To ensure that bending of the stiffened panel is kept to a minimum, the distance between the edge of the plate and the end of the stiffeners was made long enough, 340 mm, so that the small amount of bending would be handled by the flexible plate edge without transferring into the stiffened portion of the test specimen. 3.2. Specimen fabrication and test program Five specimens were fabricated to be tested. Fig. 5 shows the dimensions for specimen S1. The length and the width of the remaining specimen is the same. Plate thickness, stiffener type, stiffener spacing, and the heat input introduced in welding the stiffeners to the plates were the variables altered in the test program. Table 1 lists the test matrix for specimens S1–S5. The plate of the specimen was high-strength low-alloy steel (ASTM designation A572). The bulb-tee stiffeners were HP 160  9 and were obtained from Premier Steel Inc., Englewood, NJ, USA. The 160 and the 9 refer to the length and the thickness of the stiffener web in millimeters, respectively. The stiffeners were designated as Grade AH36 ship steel. The angle stiffeners were L101  76  8 made with corrosion resistant high strength low alloy material (ASTM designation A588).

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228.6 mm

2438.4mm

609.6 mm

W 14 x 132

203.2 mm

76.2 mm W 27 x 84 (Typ. x 4)

480 KN actuator

Stiff. Plate (Specimen)

W 12 x 50 (Typ. x 4)

4902.2mm

W 14 x 132

L 8 x 8 x 1 1/8

W 27 x 84 (Typ. x 4) Fig. 4. CAD drawings for the side view of the test frame.

3.3. Testing parameters The type of loading a structure can experience is either load control or displacement control or a mixture of both. Load control occurs when the applied load does not decrease as a result of an increasing flexibility of the system. Ship structures being exposed to waves will always experience the same load regardless of the length of the crack, or the response

ARTICLE IN PRESS 381 mm

381 mm

381 mm

240 mm

1626 mm Bulb Tee Stiffener HP 160 x 9 (Typ. x 4)

50.8 mm

340 mm

76.2 mm

3431 mm

50.8 mm

340 mm

240 mm

273

76.2 mm

H.N. Mahmoud, R.J. Dexter / Marine Structures 18 (2005) 265–288

SPECIMEN #2. 13 mm, 381 mm o. c. SPECIMEN #3. 13 mm, 381 mm o. c. VIEW “A” –“A”

SPECIMEN #4. 13 mm, 305 mm o. c. SPECIMEN #5. 9 mm, 381 mm o. c.

Fig. 5. CAD drawings of specimen S1.

of the structure. In other words, a reduction in net section will make the structure increase its displacement, but the applied loading remains the same. On a local scale, the loading experienced by the structure is displacement control. This type exists when adjacent structural members limit the displacement of a cracked member.

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274 Table 1 Test matrix Specimen #

Plate thickness (mm)

Type of stiffener

Stiffener spacing (mm)

Heat input

S1a S2a S3a S4a S5a

13 13 13 13 9

Bulb tee Bulb tee Bulb tee 101  76  8 angle Bulb tee

381 381 305 381 381

Mediumb Highb Mediumb Medium Medium

a

Plate in all specimens were fabricated from A572 (f y ¼ 50 ksi). Residual stresses were measured on these three specimens.

b

The adjacent members simply become more stressed while restraining the cracked specimen from displacement. Structurally redundant ships exhibit a great deal of displacement control behavior due to numerous load paths inherent in the cellular structure. As the crack gets bigger, the cracked component or member loses its ability to carry load and transfers or sheds the load to an un-cracked member. This phenomenon is known as load shedding. All specimens were tested in tension with a load ratio (smin/smax) of 0.2, and frequency of 0.7 Hz. These experiments were conducted under load control. The temperature was held constant at approximately 25 1C (laboratory temperature). Temperature may play a significant role when considering the fracture toughness of the material, which in return will influence the fracture behavior of the plate. However, thin structural steel plates typically meet minimum fracture toughness requirements, which are specified at temperatures much below the normal laboratory temperature. For AH36 steel the minimum Charpy V-Notch Impact required is 34 J at 0 1C, and for EH36 the minimum requirement is 34 J at 40 1C [18]. Changing the temperature of the specimen between these values would have required special equipment that was not available. The effect of temperature on the fatigue behavior is negligible. Therefore, it was felt that ignoring the temperature effect is reasonable, as the purpose of the research was to investigate crack propagation and not fracture. The length of the introduced initial cracks was kept constant (305 mm) among the specimens. One specimen was welded with a high heat input since it was believed that the high magnitude of residual stresses induced by welding the stiffeners could have the greatest influence on the growth rate of the crack. However, the measured residual stresses were not substantially different. 3.4. Test procedures and initial measurements (base line) The first specimen to be tested was specimen S1 (refer to Table 1 for details). This specimen was used for the shake down processes to assure adequate performance of the test frame. Initially the load was cycled such that the stress range in the specimen was 36 MPa with a load ratio of 0.1. A hole was drilled in the center (mid-length and mid-width) of the plate and a saw cut was made to introduce a 75 mm long crack. A weld bead was placed at each crack tip to accelerate the initiation of a fatigue crack. After undergoing a total of 490,000 cycles, a

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South Side

Extension of initial crack

275

North Side

Extension of initial crack

Initial crack (Approximately 75 mm)

Fig. 6. Initial crack introduced in specimen S1.

fatigue crack still had yet to initiate, and the length of the initial crack was increased to a total of 180 mm and bigger size weld beads were placed at the end of each crack tip (Fig. 6). Although this crack began to propagate, after about 900,000 cycles the crack had propagated away from the initial weld bead (which created tensile residual stress) and was no longer growing. The initial crack was increased to a length of 305 mm and the stress range was increased to 55 MPa, after which the fatigue crack propagated until failure of the specimen. The remaining specimens were tested with 305 mm initial cracks with weld beads at the tips and a stress range of 55 MPa and load ratio of 0.2. To distinguish between the cracks propagating on each side of the hole, the crack propagating to the right of the hole when looking at the stiffened side of the specimen is referred to as the ‘‘North side’’ crack, while the one to the left of the hole is referred to as the ‘‘South side’’ crack.

4. General experimental results Basic data of half-crack extension versus number of cycles were obtained as the cracks propagated in the stiffened panels. Crack length was measured with a fine scale. The

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700

Crack extension, (mm)

600

CCT (FEA) Stiffener, (TYP.)

500

S1 North S1 South S2 North S2 South S4 North S4 South S5 North S5 South

400 300 200 100 0 0

100000

200000

300000 400000 500000 Number of cycles

600000

700000

800000

Fig. 7. Crack extension versus cycles for S1, S2, S4 and S5, and a specimen with no stiffeners attached (using finite-element analysis).

location of the crack tip was enhanced with a red dye that penetrates through the crack when it opens and comes back out as the crack closes. Fig. 7 shows crack extension versus number of cycles for specimens S1, S2, S4 and S5. For reference, the predicted crack propagation of a center-cracked tension (CCT) plate with no stiffeners is also shown. The predicted CCT data was obtained using FE analysis. The data for specimen S3 is shown separately in Fig. 8 as it had different stiffener spacing than the rest of the specimens. There is very good agreement between the results of the north and south crack in these experiments. However, there is considerable variation between specimens. Specimen S2 and S4 agree very well with each other, which is believed to be coincidence. Note that S2 and S4 have different stiffener types, so this indicates that the type of stiffener is not an important variable. It is not known why S1 and S5 took longer to propagate than S2 and S4, but crack propagation is highly variable as discussed previously. The results could be dependent on a number of factors that were not controlled, such as the sequence in which the stiffeners were welded to the plates. Such variations would be expected to occur in actual ship construction as well. Therefore, an upper bound crack propagation model should be used. As can be seen in Figs. 7 and 8, when testing a single plate with no redundancy, the number of cycles it took to propagate the crack between the stiffeners was twice as many as in the case of the CCT specimen with no stiffeners. The same observation was made by Dexter and Pilarski [19] when redundant box sections were tested. Therefore, redundancy had a small effect on the crack growth rate. It is important to keep in mind, however, that while it has little effect on the crack propagation rate, having a redundant structure could be extremely beneficial in keeping the ship intact and preventing break up of the ship as large cracks develop. It is important to mention that the graph for the CCT specimen in Figs. 7 and 8 was obtained from FE analysis. The existence of residual stresses from the welding process increased the number of cycles by a factor of two.

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800

700 CCT (FEA)

Crack extension, (mm)

600

500

400 Experiment North 300

Stiffener, (TYP.)

200 Experiment South 100

0 0

100000

200000

300000 Number of cycles

400000

500000

600000

Fig. 8. Crack extension versus cycles for S3 and a CCT specimen with no stiffeners attached (using finite-element analysis).

Table 2 lists the crack length in the plates and up the stiffeners for all specimens as well as the remaining sections. Stiffener 1 in the table refers to the first stiffener the crack encounters as it grows (inner stiffener), and Stiffener 2 refers to the second stiffener that the crack propagates through (outer stiffener). Fig. 9 shows the final crack appearance in specimen S1 (after test was terminated) was the crack propagating in the plate and in the web of the second stiffener. 5. Residual stresses measurements The measurement of residual stresses was done on specimens S1–S3 using the sectioning method. S1 was sectioned using 41 coupons with a nominal gage length of 254 mm. Four coupons from either side of each stiffener were taken with a width of 12 mm. Three additional coupons with a width of 37 mm were taken from the region between stiffeners. S2 and S3 were sectioned using a lesser number of coupons with a nominal gage length of 100 mm. The method of sectioning was chosen to be the most economical and convenient method for measuring residual stress. A comparison study between various techniques of residual stresses measurements showed that the sectioning method yields the most accurate results with standard deviation of 710 MPa, when measured in a normal case [20]. Before sectioning the plate, the gage points were drilled through the thickness of the plate at the center of the sections marked on the plate using a 3 mm diameter drill bit. It

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Table 2 Final crack extension in plate and stiffeners Specimen

S1

S2

S3

S4

S5

Cycles count Half-crack (Plate north)

521,804 543 mm & 109.8 mm remain 144 mm & 16 mm remain

547,126 652.8 mm & 0 mm remain

498,913 412.8 mm & 160 mm remain

757,652 598 mm & 54 mm remain

Half-crack (stiff. 1 north)

635,306 420 mm & 232.8 mm remain 145 mm & 15 mm remain

150 mm & 10 mm remain

92 mm & 11 mm remain

Half-crack (stiff. 2 north)

34 mm & 126 mm remain

87 mm & 73 mm remain

145 mm & 15 mm remain

10 mm & 91 mm remain

108 mm, then severed at 523,221 cycles 110 mm & 50 mm remain

Half-crack (plate south)

558 mm & 94.8 mm remain

478.8 mm & 174 mm remain

427.8 mm & 225 mm remain

432 mm & 212 mm remain

Half-crack (stiff. 1 north)

149 mm & 11 mm remain

421 mm & 231.8 mm remain 142 mm & 18 mm remain

152 mm & 8 mm remain

92 mm & 11 mm remain

Half-crack (stiff. 2 south)

90 mm & 70 mm remain

10 mm & 150 mm remain

115 mm & 45 mm remain

23 mm & 78 mm remain

148 mm, then severed at 523,221 cycles 30 mm & 130 mm remain

was believed that the inherent bending in the plate would have an effect on the measured values, and could possibly result in overestimating the residual stresses. Therefore, drilling a hole all the way through the plate allowed for the measurements to be taken from each side of the plate at the exact same location on the extracted section. A digital Whitmore gage with accuracy to 0.001 mm was used to measure the distance between the gage points. Three readings were taken on each side of the plate and the average of each three was used. The average of both sides of the plate (the stiffener side and the non-stiffener side) was used to obtain the residual stresses. After recording the initial readings, the plate was sectioned into coupons. The coupons were extracted from the larger section using a band saw that was cooled with a steady flow of cutting fluid. The coolant fluid was applied to prevent any heat input into the plate that might be introduced through the cutting process. After removing the coupons, the final readings were obtained by averaging three readings for each side and then averaging the results of both sides. Fig. 10 shows the sectioned portion of specimen S1. As expected, some scatter existed in the measured values of the residual stresses. In specimen S1, the values of the measured compressive residual stresses were between 50 and 100 MPa. For equilibrium to take place, assuming that the tensile residual stress is 350 MPa at the weld lines, a constant compressive residual stress value of approximately 75 MPa must be the actual value present in the plate. This assumption was verified in the FE analysis as it shows that an assumed value of 75 MPa would yield a very similar behavior for crack propagation as was seen in the experiment. More scatter in the measured values was observed in specimen S2 than in S1. It is unclear why there is more scatter in the measured values in S2 considering that the steps taken were identical. Faulkner’s model [17] for residual stress distribution was utilized as a simple representation of the actual residual stress in the specimens. This model presents the

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Approximately 90 mm

Fig. 9. Final crack appearance in the plate and the second stiffener (south side) of specimen S1.

tensile regions around the stiffeners as rectangular shapes with a base width proportional to the plate thickness. The rectangular width typical of as-built ship structures ranges from 3.5 to 4 times the plate width, while values between 3 and 3.5 are more typical of ships after shakedown. Figs. 11 and 12 show the typical residual stresses measured in the plates for specimens S1 and S2, and S3, respectively. Even though the values of the measured residual stresses are not in equilibrium (i.e. the area under the compressive zone is not equal to the area under the tension zone), the values seem to be symmetric. The scatter in the measured values was expected, but not accounted for in the FE analysis. 6. FE analyses FE analysis was used as a tool to investigate the crack propagation in the stiffened panel. Generally, static analysis of stress intensity factors is used to determine the crack propagation rate by comparing the critical value of the stress intensity factor KIC to the Kvalue associated with a specific crack length. The static analysis could be done many times to obtain K for various crack sizes. These values are then used to determine the number of cycles required to propagate the crack a distance da. For a specified direction cosine of the

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254 mm x 37 mm coupon with drilled gage points

Fig. 10. Sectioned coupons used for measuring residual stress distributions in S1.

crack as well as the node number at the crack tip, the J-Integral values were computed, and then converted to K values using Eq (3). The J-Integral is a fracture characterizing parameter equivalent to energy release rate in non-linear elastic materials. It was presented by Rice [21] as a path independent contour integral for analysis of cracks. Mathematically, J is obtained by evaluating an integral around a path containing the crack tip [22]: pffiffiffiffiffiffi K ¼ JE , (3) where J is the J-Integral and E is the Young’s Modulus. The values of Kapp, max, Kapp, min, and Kres previously discussed were found by running the analysis for every load case separately, then using superposition to calculate the value of DKeff to be used in the Paris-law equation. When running these analyses, one must be careful not to apply Ds to the plate and find the DJ, then convert the value to DK, as J and K do not have a linear relationship. 6.1. Numerical model development A 3-D analysis was performed using ‘‘ABAQUS’’ Software, which was available through the University of Minnesota Super Computer Institute. Because of symmetry,

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500 Stiffener, (TYP) 400

Residual stress, (MPa)

300

200

100

0

-100 S1

S2

3.5 x η

-200 -812 -712 -612 -512 -412 -312 -212 -112 -12

88

188

288

388

488

588

688

788

488

588

688

788

Distance from center of plate Fig. 11. Residual stresses measurements in S1 and S2.

400 Stiffener, (TYP)

Residual stress, (MPa)

300

200

100

0

-100

-200 -812 -712 -612 -512 -412 -312 -212 -112 -12 88 188 Distance from center of plate

288

Fig. 12. Residual stresses measured in S3.

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U1 = Prescribed value

U2 = 0

3 1 U1 = 0

2

Fig. 13. Typical mesh and boundary conditions.

only 1/4 of the plate was modeled, which helped in reducing the time required for analysis significantly. As the measured residual stresses indicated, high stress gradient between the region of compressive residual stresses and tensile residual stresses existed in all panels (as suggested by Faulkner [17]). To better capture the stress gradient, it was felt that using eight-node quadrilateral shell elements with reduced integration would be appropriate. A typical mesh size used for the analysis was 5 mm at and around the crack face region. Fig. 13 shows a typical model used in the analyses. 6.2. Residual stresses modeling The stresses were introduced in the plate by specifying initial tensile stresses in a region equal to Z*tplate, where Z is equal to 3.5. This value represents the tension block suggested by the Faulkner model. The stresses were specified in the desired local direction for the selected elements. The compressive residual stresses were then generated in the analysis to satisfy equilibrium. Because there was quite a significant scatter in the measured values of residual stresses, it was desirable to see the effect of the magnitude of these stresses on crack propagation. This was done by conducting the analysis three different times using compressive residual stresses values of 50, 75, and 100 MPa. These values were chosen as they approximately represent the minimum, average, and maximum measured values in specimen S1. The values of Kres were calculated the same way as Kapp. The nodes were released to propagate the crack to the desired length, allowing for the redistribution of residual stresses. The effect of the magnitude of the compressive residual stresses on DKeff, could easily be seen in Fig. 14. If the magnitude of residual stresses was zero, then both lines representing DKeff and DKapp should merge (i.e. no gap). In the regions of high compressive residual stresses, crack closure could occur causing an overlap of the crack. Nussbaumer [4] used gap elements to overcome this issue. Dexter and

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120

Stiffener, (TYP.)

100

∆K, MPa * (m)0.5

80

Applied DK

60

40 Effective DK 20

0 0

100

200

300

400 500 Half crack length, (mm)

600

700

800

900

Fig. 14. DKapp and DKeff for compressive residual stresses of 100 MPa.

Pilarski [5] did a case study on the use, versus non-use of gap elements and reported that using gap elements introduces geometric non-linearity, which conflicts with the use of LEFM. They also found that using gap elements yields similar results as if they were not used. Furthermore, Kres derived from a gap element analysis is a little higher than the Kres, resulting in a conservative value of DKeff. 6.3. Comparison between FE and experimental results The FE results varied based on the values used for the Paris equation parameter C, as well as the magnitude of the residual stresses. A parametric FE analyses study with conducted using different values of the parameter C as well as different magnitude of residual stresses to find the most appropriate combination, which well represents the actual pffiffiffiffi experiments behavior. Fig. 15 shows that a combination of a C-value of 9.5E12 MPa m and a compressive stress value of 75 MPa were the most appropriate values to be used for representing the actual propagation rate in the plate. 7. Analytical program For a stiffened panel, the analytic program is sensitive to the type and magnitude of the residual stress distribution. With the FE model, reasonable but conservative results were obtained with the compressive residual stress equal to 75 MPa between stiffeners. Fig. 16 shows the results of the analytical model for compressive residual stress of 75 MPa

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700 Experiment North 600 75 MPa C=9.5E-12 Crack extension, (mm)

500

50 MPa C=9.5E-12

400 F. E, KCCT C = 9.5E-12

300

200 Experiment South

100 MPa C=9.5E-12

Stiffener, TYP.

100

0 0

100000 200000 300000 400000 500000 600000 700000 800000 900000 1000000 Number of Cycles

Fig. 15. Comparison between finite-element analysis and experimental results for crack extension versus cycles using compressive residual stresses of 50, 75, and 100 MPa (specimen S2).

700

Analytical, ∆σ = 55 MPa, 0.25 µ

600

Analytical, ∆σ = 55 MPa,

Crack Extension, (mm)

500

400

S1 North S1 South S2 North S2 South CCT S4 North S4 South S5 North S5 South

CCT, ∆σ = 55 MPa,

300

Stiff. (TYP.)

200 Analytical, ∆σ = 42 MPa

100

0 0

200000

400000 600000 Number of Cycles

800000

1000000

Fig. 16. Results of the analytical model for compressive residual stress of 75 MPa compared to the four experiments with the same stiffener spacing.

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700 25.0 µ, restraint, no residual 600 CCT (FEA)

Crack Extension, (mm)

500

400 0.25 µ, no restraint, residual

300

Stiff. (TYP.)

200

0.25 µ, restraint, residual

100

0 0

50000

100000 150000 200000 250000 300000 350000 400000 450000 500000 Number of Cycles

Fig. 17. Result of the analytical model corrected for shear lag compared to equivalent analyses with no restraint effect and no residual stress effect.

compared to the four experiments with the same stiffener spacing. Notice the very good agreement between the shape of the curves for the analytical model compared to shape of the experimental curves. With the analytical model, the relative magnitude of several different effects can be determined. Fig. 17 shows the result of the analytical model corrected for shear lag compared to equivalent analyses with no restraint effect and no residual stress effect. It can be seen that both effects are about equally important. The case with no restraint effect tracks the CCT result until the residual stress kicks in at crack extension of about 200 mm. Both cases with restraint have a delay in the crack propagation at the outset but then eventually becoming more or less parallel with the CCT result. Then the case with restraint and residual stresses begins another delay due to residual stress. The measured stress range in the plate in these tests was 55 MPa, however the stress was not uniform and the stress range in the stiffeners decreased with distance away from the plate but averaged less than 25% of 55 MPa. Therefore, when the model was run with 55 MPa as the stress range, the results are too conservative. The reason the results are too conservative is that the stiffener force, which is overestimated at 55 MPa, acts to increase stress-intensity factor from the splitting forces after the stiffener is severed. The total load range divided by the cross-section area gives a stress range of 42 MPa. This is less than the 55 MPa because the 42 MPa is evenly distributed. When the analytical

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model was run with this lower stress range, the results are not conservative. This is because the stress range in the plate is much higher (55 MPa) in reality. As an attempt to model the shear lag in the stiffeners with the analytical model, the m (stiffener to plate stiffness ratio) was multiplied by 0.25, to reflect average stiffener stress range 25% of the stress range in the plate. 8. Conclusion Five experiments were conducted to investigate fatigue crack propagation in welded stiffened panels, which are typical of ship structures. The specimens varied in the spacing, geometry, and heat input used for welding of the stiffeners. Linear-elastic FE models of the specimens were developed to calculate the range in the stress intensity factor, which was then used to estimate the crack propagation rate. Residual stresses were incorporated in the model as initial stresses. The model was validated by comparing the calculated crack propagation with the experimental results. The following conclusions were reached: 1. The crack propagation remained stable despite crack lengths up to 750 mm and peak stress of 70 MPa. 2. There is a substantial reduction in the crack propagation rate in stiffened plates relative to what would be expected in a plate without stiffeners. The cycles to propagate one stiffener spacing may increase by a factor of 2–4 above what would be predicted for a center-cracked plate. 3. Crack propagation rate in the specimens was similar to the propagation rate observed in highly redundant box sections loaded in bending tested in a previous project, suggesting that redundancy and stress gradient in the box section tested previously had little effect on the propagation rate. 4. Two factors cause the reduction in the crack propagation rate in a stiffened panel: the restraint effect and the compressive residual stress between stiffeners arising from the welding process. These two factors are about equally important in these experiments. 5. Measured residual stress showed high variability in magnitude with a consistent pattern of tensile residual stress adjacent to the stiffeners and compressive residual stress between the stiffeners. The pattern is consistent with past studies of residual stress in welded stiffened panels. 6. The spacing of the stiffeners has an obvious effect on the reduction in crack propagation rate as well as the overall number of cycles to failure. The magnitude of the compressive residual stresses increases as the spacing between the stiffeners decreases. 7. The crack propagation rate up in the stiffener’s web has the same rate as it continues to propagate in the plate pass the stiffener. 8. The analysis using the FE method to calculate the range in stress intensity factor, followed by using Paris’s Law to predict the crack propagation, gave good agreement with the experiments provided that reasonable values for the parameters of the crack growth rate equation were chosen. 9. Simple spreadsheet analysis gave as good agreement with the experimental data as the FE analyses at much lower effort. It is recommended that no finite-width correction be used with this analytical model.

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Acknowledgements The first author would like to dedicate this paper to the soul of Dr. Robert Dexter from the Civil Engineering Department at the University of Minnesota, whose sudden death on November 16, 2004 brought a tragic end to a great researcher, engineer, educator, and before all a very dear friend. The study was conducted at the Structures Laboratory of the Department of Civil Engineering at the University of Minnesota. The authors wish to especially thank Paul Bergson, the manager of the Structures laboratory, for his ideas, support, and work in completion of the difficult experimental work. This work was made possible by the generous contribution of the US Coast Guard R&D, particularly Mr. Robert Sedat. The project was administered by the Ship Structure Committee through a contract to J.J. McMullen Associates (JJMA). The authors are grateful for the support and for the guidance of Raymond Kramer and Peter Fontneau of JJMA and the Project Technical Committee, particularly David Stredulinsky of Defence R&D Canada. The authors are also thankful for the valuable advices, which were provided by Harold Reemsnyder, Bethlehem, PA, Sreekanta Das, Defence R&D Canada, Rong Huang, Danville, CA, John Sumpter, Qinetiq, Rosyth, UK. References [1] www.imo.org/newsroom/mainframe.asp?topic_id ¼ 72. [2] Tanker Structure Co-operative Forum. Int’l chamber of shipping and oil companies int’l marine forum, guidelines for the inspection and maintenance of double hull tanker structures, London, England; 1995. [3] Tanker Structure Co-operative Forum. Int’l chamber of shipping and oil companies int’l marine forum, guidance manual for the inspection and condition assessment of tanker structures, London, England, 1986. [4] Nussbaumer A. Propagation of long fatigue cracks in multi-cellular box beams. Ph.D. dissertation, Lehigh University, 1993. [5] Dexter RJ, Pilarski PJ. Crack propagation in welded stiffened panels. J Constr Steel 2002;58:1081–102. [6] Anderson TL. Fracture mechanics. Boca Raton, FL: CRC Press, LLC; 1995. [7] Rushton PA. An experimental and numerical investigation into fatigue crack propagation of 350 WT steel subjected to semi-random cyclic loading. Report no. DREA CR 2000-154, Defense Research Establishment Atlantic, Canada, March 2001. [8] Paris P, Erdogan F. A critical analysis of crack propagation laws. J Basic Eng 1963;85:528–34. [9] Kober GR, Dexter RJ, Kaufmann EJ, Yen BT. The effect of welding discontinuities on the variability of fatigue life. ASTM Spec Tech Publ 1995;1220:553. [10] Barsom JM, Rolfe ST. Fracture and fatigue control in structures. Englewood Cliffs, NJ: Prentice-Hall; 1987. [11] BS 7910. Guide on methods for assessing the acceptability of flaws in metallic structures. London: British Standards Institute; 1999. [12] Fisher JW, Dexterm RJ, Roberts R, Yen BT, Decorges G, Pessiki SP, Nussbaumerm AC, Tarquinio JE, Kober GR, Gentilcore ML, Derrah SM. Structural failure modes: final report-development of advanced double hull concepts. Final report for cooperative agreement N00014-91-CA-001, vol. 3a, Lehigh University, Bethlehem, USA, March 1993. [13] BS6493. Fatigue design and assessment for steel structures. London, UK: British Standards Institute; 1993. [14] Poe CC. The effect of riveted and uniformly spaced stringers on the stress intensity factor of a cracked sheet. Master of Science dissertation, Virginia Polytechnic Institute, 1969. [15] Poe CC. Fatigue crack propagation in stiffened panels. Damage tolerance in aircraft structures. American Society for Testing and Materials 1971;ASTM STP 486:79–97. [16] Thayamballi AK. Reliability of ship hull in the fracture and fatigue modes of failure. Ph.D. dissertation in Engineering, University of California, 1983. [17] Faulkner D. A review of effective plating for use in the analysis of stiffened plating in bending and compression. J Ship Res 1975;19:1–17.

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[18] Standard Specification for Structural Steel Ships, Designation: A131/A 131M-01, August, 2001. [19] Dexter RJ, Pilarski PJ. Effect of welded stiffeners on fatigue crack growth rate. Report no. 413, Ship Structure Committee, Washington, DC, August 2000. [20] Lu J. Handbook of measurements of residual stresses. Society For Experimental Mechanics, Inc.; 1996. p. 228–30. [21] Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 1986;35:379–86. [22] Cook RD, Malkus DS, Plesha ME. Concepts and applications of finite element analysis. New York: Wiley; 1989.